Mass-Radius Relation for Rocky Planets based on PREM
Li Zeng1,a , Dimitar Sasselov2,b , and Stein Jacobsen1,c
arXiv:1512.08827v2 [astro-ph.EP] 4 Feb 2016
1
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138
2
Astronomy Department, Harvard University, Cambridge, MA 02138
a
b
c
[email protected]
[email protected]
[email protected]
ABSTRACT
Several small dense exoplanets are now known, inviting comparisons to Earth
and Venus. Such comparisons require translating their masses and sizes to composition models of evolved multi-layer-interior planets. Such theoretical models
rely on our understanding of the Earth’s interior, as well as independently derived equations of state (EOS), but have so far not involved direct extrapolations
from Earth’s seismic model -PREM. In order to facilitate more detailed compositional comparisons between small exoplanets and the Earth, we derive here a
semi-empirical mass-radius relation for two-layer rocky planets based on PREM:
R
= (1.07 − 0.21 · CMF) · ( MM⊕ )1/3.7 , where CMF stands for Core Mass Fraction.
R⊕
It is applicable to 1∼8 M⊕ and CMF of 0.0∼0.4. Applying this formula to Earth
and Venus and several known small exoplanets with radii and masses measured
to better than ∼30% precision gives a CMF fit of 0.26 ± 0.07.
Subject headings: Extrasolar planets, interiors, mass-radius relation, core mass
fraction
1.
Introduction
A first step in deriving the compositional diversity of small rocky planets was accomplished recently by (Dressing et al. 2015), who added Kepler-93b to the list of a dozen or
so small exoplanets with radii and masses measured to better than ∼30% precision. With
exquisite sizes (mostly from Kepler light curves) and huge follow-up effort (Dressing et al.
2015; Pepe et al. 2013; Batalha et al. 2011; Ballard et al. 2013; Carter et al. 2012; Hatzes et al.
–2–
2011; Berta-Thompson et al. 2015) the mass-radius diagram is finally amenable to some detailed comparisons with theory in the 1 to 10 M⊕ range. Closer to the mass and size of Earth,
the rocky planets known to-date seem to exhibit an unexpectedly tight compositional correlation. Is this correlation really shared with Earth and Venus? If so, to what level of precision
and under what assumptions? To begin answering such questions we must first acknowledge
that models of the interior structure and composition of rocky exoplanets in the Super-Earth
domain are based largely on experience (and extrapolations) from the models of the rocky
solar system planets, and mostly - the Earth (Valencia et al. 2006, 2007b,a; Fortney et al.
2007; Seager et al. 2007; Zeng & Seager 2008; Grasset et al. 2009; Zeng & Sasselov 2013).
Here instead of using simple shell models based on EOSs of minerals and metals, we take
a different approach and derive semi-empirical EOSs based on the well-constrained seismic
model of the Earth.
2.
Equation of State (EOS)
On one hand, in several previous models of solid exoplanets (Zeng & Sasselov 2013;
Zeng & Seager 2008; Seager et al. 2007), cores and mantles of solid exoplanets are modeled as
pure ǫ-Fe-solid and Mg-perovskite/post-perovskite respectively. On the other hand, we know
the actual density variation inside Earth through measurements of seismic wave velocities.
This seismically derived density model is widely known as PREM (Preliminary Reference
Earth Model (Dziewonski & Anderson 1981)). The differences between the two approaches,
primarily (1) the liquid outer core less dense than pure ǫ-Fe solid and (2) the upper mantle
less dense than the extrapolation of lower mantle, can cause differences in the mass-radius
relations derived.
There are several assumptions made when we extrapolate PREM. We assume the upper
mantle to lower mantle phase transition occurs at the same pressure as the Earth’s interior.
It is a pressure-driven phase transition, so temperature effect is secondary and thus ignored.
The Thomas-Fermi-Dirac Model modified with Correlation Energy (Salpeter & Zapolsky
1967) (abbreviated as TFD from now on) serves as a lower bound for the density of material considered. Any properly-behaving EOS should asymptotically approach TFD above
∼1 TPa, as at such high pressures the electron degeneracy pressure dominates while the
detailed chemical and crystal structures of the material become less important.
We will show that the Birch-Murnaghan 2nd order EOS (abbreviated as BM2 from
now on) (Birch 1952) provides a fairly decent description of how material is compressed in
Earth’s interior for both core (good up to 12 TPa where it asymptotically approaches TFD)
–3–
and mantle (good up to 3.5 TPa where it asymptotically approaches TFD). Those pressures
roughly correspond to the central pressure and core-mantle boundary pressure respectively
of a 30 M⊕ rocky planet with CMF=0.3.
3
P = · K0
2
"
ρ
ρ0
37
−
ρ
ρ0
53 #
(1)
The fit to lower mantle, outer core and inner core PREM gives:
• Lower Mantle: ρ0 = 3.98 g/cc, K0 = 206 GPa, error∼ 1% in ρ.
• Outer Core: ρ0 = 7.05 g/cc, K0 = 201 GPa, error∼ 1% in ρ.
• Inner Core: ρ0 = 7.85 g/cc, K0 = 255 GPa, error∼ 0.01% in ρ.
The uncertainties of the fit are similar to the intrinsic uncertainty of PREM in ρ of ∼ 1%
(See (Ricolleau et al. 2009) and (Ritsema et al. 2011) and Figure 3 of (Huang et al. 2011)).
2.1.
PREM-extrapolated EOS for Mantle
Inside Earth, the density jump from upper mantle to lower mantle is 10% from 4 g/cc to
4.4 g/cc at 23.83 GPa. Earth’s upper mantle consists of complex phases of Mg-silicates, including various polymorphs of olivine: (α) olivine, (β) wadsleyite, and (γ) ringwoodite (Bina
2003). The upper mantle to lower mantle transition occurs at 670 km depth in Earth (about
10.5% Earth radius). For more massive planets like Kepler-93b, this transition occurs at
shallower depth (∼ 3% radius of Kepler-93b) for the same pressure.
Fig. 1 shows the comparison of different EOS of Mg-silicates.
Interestingly, the TFD beyond 1 TPa for MgO, SiO2 , MgSiO3 , and Mg2 SiO4 are almost
identical as they all have average atomic weight A=20 and average atomic charge Z=10.
This coincidence simplifies the EOS of Mg-silicates, as it indicates that Mg/Si ratio does not
matter towards the high-pressure end. It only matters towards the low-pressure end, which
is captured in our EOS by using the PREM density variation in the upper-mantle pressure
range.
According to Fig. 1, except the prominent density jump of 10% at the upper-lower
mantle boundary, the other high-pressure phase transitions are only ∼ 1% level in density
and thus insignificant in mass-radius calculation (Unterborn et al. 2015). Therefore, in this
paper we adopt the EOS of Mg-silicates mantle as follows:
–4–
Fig. 1.— Orange points: PREM density of Earth’s mantle, excerpt from Appendix G
of (Stacey & Davis 2008). Pink curve: TFD for A=20 and Z=10. Black curve: BM2 fit
to lower-mantle PREM (ρ0 = 3.98 g/cc, K0 = 206 GPa). Green curve: including lowermantle phase transitions and pv-ppv phase transition at 122 GPa (Zeng & Sasselov 2013;
Caracas & Cohen 2008; Spera et al. 2006) and further dissociation of ppv at 0.9 and 2.1
TPa (Umemoto & Wentzcovitch 2011; Wu et al. 2011).
–5–
• 0 GPa - 23.83 GPa: linear interpolation of the lower mantle density according to the
Appendix G of (Stacey & Davis 2008), which is taken from PREM. i.e. follow the
green curve in Fig. 1.
• 23.83 GPa - 3.5 TPa: BM2 EOS with ρ0 = 3.98 g/cc, K0 = 206 GPa, (error∼ 1% in
density). i.e. follow the black curve in Fig. 1.
• >3.5 TPa: TFD EOS of MgSiO3 calculated using method in (Salpeter & Zapolsky
1967). i.e. follow the pink curve in Fig. 1.
2.2.
PREM-extrapolated EOS for Core
Fig. 2 shows the comparison of different EOS of the Core.
In Earth, the density jump from liquid outer core to solid inner core is 5% from 12.2
g/cc to 12.8 g/cc at 328.85 GPa. The Rydberg EOS of pure solid ǫ-Fe phase, which is
experimentally determined from static compression up to 205 GPa (Dewaele et al. 2006;
Wagner et al. 2012), is too high of density (cyan curve in Fig. 2) in the pressure region
relevant to Earth, even including the temperature effect (green curve in Fig. 2), even for
the solid inner core. This is likely due to the presence of several weight percent of one or
more light elements (McDonough 2014). The lighter elements could be S, Si, O, C, or a
combination of them (Badro et al. 2007; Fischer et al. 2012; Alfè et al. 2002; Poirier 1994;
Stixrude et al. 1997; Anderson & Ahrens 1994), but no agreement is reached as to which of
these elements are most important. The Rydberg EOS with or without temperature behaves
poorly for extrapolation above∼1TPa as it overshoots and crosses over the TFD EOS, which
serves as a lower bound for the density of Fe.
As such, we choose to only extrapolate the liquid outer core BM2 EOS to higher pressure
until it asymptotically approaches the TFD EOS. The solid inner core is likely a secondary
feature resulting from the crystallization of the liquid core. It currently comprises only a
small fraction (≈ 3%) of the total mass of the Earth, and in more massive planets, the focus
of this study, solid inner core may not exist at all due to higher heat content and slower
cooling rate. BM2 EOS behaves nicely as it asymptotically approaches the TFD EOS at
vert high pressure (∼12 TPa).
Therefore, in this paper we adopt the EOS of Fe core as follows:
• 0 GPa - 12 TPa: BM2 EOS with ρ0 = 7.05 g/cc, K0 = 201 GPa, (error∼ 1% in
density). i.e. follow the black curve in Fig. 2.
–6–
Fig. 2.— Orange points: PREM density of Earth’s core, excerpt from Appendix G
of (Stacey & Davis 2008). Pink curve: TFD of iron (A=55.845 and Z=26). Black curve:
BM2 fit to outer-core PREM (ρ0 = 7.05 g/cc, K0 = 201 GPa). Gray dashed curve: BM2 fit to
inner-core PREM (ρ0 = 7.85 g/cc, K0 = 255 GPa). Cyan curve: Rydberg ǫ-Fe EOS at 300K
isotherm according to (Dewaele et al. 2006). Green curve: Rydberg ǫ-Fe EOS (Dewaele et al.
2006) with T-P profile (small inlet plot) of rocky planet core interpolated from Wagner et al.
(2012).
–7–
• >12 TPa: TFD EOS of Fe (A=55.845 and Z=26) calculated using method in (Salpeter & Zapolsky
1967). i.e. follow the pink curve in Fig. 2.
3.
Mass-Radius Relation
Dressing et al. (2015) points out a tight mass-radius relation of solid exoplanets between
2 and 5 M⊕ from the comparison of Earth, Venus, and several dense exoplanets by using the
mass-radius contours described in (Zeng & Sasselov 2013). However, Dressing et al. (2015)’s
CMF fit of 17% is much lower than that of Earth or Venus. Here we show that using the
PREM-extrapolated EOS renders a better fit of CMF in agreement with that of Earth and
Venus (See Fig. 3).
There might be a selection bias towards higher density planets by selecting only the
planets with better than ∼30% mass measurement accuracy. However, as already pointed
out by Dressing et al. (2015), the low mass planets with very low densities like Kepler-11b
and Kepler-138d do not detract from the conclusion that all the rocky analogs of the Earth
obey a single mass-radius relation. The degree to which this is true will be tested by more
precise mass measurements.
The planets in Fig. 3 are tabulated in Table 1.
Table 1: CMF of Planets
Planet
M(M⊕ )
Earth1
1
Venus1
0.815
K-10b
3.33 ± 0.49 (Dumusque et al. 2014)
+0.33
K-36b
4.45−0.27
(Carter et al. 2012)
+0.38
(Pepe et al. 2013)
K-78b
1.86−0.25
K-93b
4.02 ± 0.68 (Dressing et al. 2015)
COROT-7b
5.74 ± 0.86 (Haywood et al. 2014)
HD219134b
4.46 ± 0.47 (Motalebi et al. 2015)
GJ1132b
1.62 ± 0.55 (Berta-Thompson et al. 2015)
R(R⊕ )
CMF
1
0.325 ± 0.001
0.9499
0.31 ± 0.01
+0.03
1.47−0.02
(Dumusque et al. 2014)
0.04 ± 0.2
1.486 ± 0.035 (Carter et al. 2012)
0.37 ± 0.14
+0.159
+0.3
(Pepe et al. 2013)
1.173−0.089
0.37−0.5
1.478 ± 0.019 (Ballard et al. 2013)
0.26 ± 0.2
1.585 ± 0.064 (Barros et al. 2014)
0.14 ± 0.3
1.606 ± 0.086 (Motalebi et al. 2015)
0.0 ± 0.3
1.16 ± 0.11 (Berta-Thompson et al. 2015)
0.25 ± 0.5
The mass-radius curves in Fig. 3 are tabulated in Table 2.
1
In the calculation, we treat the errors in CMF of Earth and Venus as ±0.2, comparable to the errors for
exoplanets considered here, so as not to bias the fit.
2
mass in Earth Mass (M⊕ = 5.9742 × 1024 kg)
3
radius in Earth Radius (R⊕ = 6.371 × 106 m)
–8–
Table 2: Mass-Radius Table
M(M⊕ )
0.125
0.25
0.5
1
2
4
8
16
32
2
100%fe
R(R⊕ )3
0.445
0.55
0.676
0.823
0.99
1.176
1.38
1.59
1.82
3.1.
50%fe
R(R⊕ )
0.523
0.645
0.789
0.961
1.164
1.4
1.66
1.94
2.25
30%fe
R(R⊕ )
0.547
0.672
0.823
1.005
1.22
1.47
1.75
2.06
2.38
25%fe
R(R⊕ )
0.553
0.679
0.832
1.016
1.23
1.49
1.77
2.08
2.42
20%fe
rock 25%h2o 50%h2o
R(R⊕ ) R(R⊕ ) R(R⊕ )
R(R⊕ )
0.558
0.58
0.649
0.697
0.685
0.711
0.793
0.851
0.84
0.872
0.969
1.039
1.026
1.067
1.182
1.27
1.25
1.3
1.44
1.54
1.5
1.57
1.74
1.85
1.79
1.88
2.07
2.21
2.11
2.22
2.45
2.61
2.45
2.58
2.85
3.04
100%h2o
R(R⊕ )
0.776
0.952
1.163
1.41
1.71
2.05
2.45
2.9
3.36
Mass-Radius Formula Fitting for Rocky Planets
The CMF-dependent mass-radius relation of rocky planets can be fit to the following
formula (for 0≤CMF≤0.4, 1M⊕ ≤ M ≤ 8M⊕ ). It agrees with the actual mass-radius curves
in Fig. 3 within ∼ 0.01 R⊕ in radius. If we use it to calculate CMF, it gives an uncertainty
of ∼ 0.02 in CMF.
R
R⊕
M
= (1.07 − 0.21 · CMF)
M⊕
1/3.7
(2)
Eq. 2 can be inverted to solve for CMF given the mass and radius:
CMF =
1
R
M 1/3.7
[1.07 − (
)/(
)
]
0.21
R⊕ M⊕
(3)
For comparison, this semi-empirical mass-radius formula is in agreement with the massradius relation of super-Earths presented in Valencia et al. (2006), which is a scaling law
of R ∝ M 0.267−0.272 . This new formula takes one step further to articulate the dependence
on CMF, which is useful in differentiating among rocky planets with different CMF (i.e.
compositions).
1
[1.07 − (1)/(1)1/3.7 ] = 0.07/0.21 = 0.33. For Kepler-93b,
For Earth, CMF⊕ = 0.21
1
CMFK93b = 0.21 [1.07 − (1.478)/(4.02)1/3.7] = 0.26. The uncertainties (δCMF) in CMF resulting from the uncertainties in mass and radius can be derived from Eq. 3:
–9–
|δCMF| ≈ 5 ×
r
|
δr 2
1
δm 2
| + ( )2 |
|
r
3.7 m
(4)
So in order to tell a 20% core mass apart from a 30% core mass, radius needs to be
measured to better than 2% level, or equivalently, mass to 6% level. For Kepler-93b, | δrr | =
0.68
0.019
= 0.013 and | δm
| = 4.02
= 0.17, so |δCMF| = 0.2. Therefore, CMFK93b = 0.26 ± 0.2.
1.478
m
Table 1 Column 4 lists the CMF estimates of these exoplanets.
Assuming this population of dense exoplanets follows the same CMF distrbution, we
then apply a weighted least-square fit to Table 1.
Denote the weighted average of CMF as CMF:
P
i
CMF = P
i
CMF
δCMF2
1
δCMF2
(5)
Denote the standard deviation of CMF as σCMF:
v
u
X
u
σCMF = t1/
i
1
δCMF2
!
(6)
Result: CMF = 0.26 and σCMF = 0.07, indicating Earth-like composition (CMF ∼ 0.3)
carries on up to at least ∼5 M⊕ .
3.2.
Discussion
This is backed up by recent studies of disintegrated planet debris in polluted white dwarf
spectra (Jura & Young 2014; Wilson et al. 2015; Xu et al. 2014). These studies show that
the accreted extrasolar planet debris generally resemble bulk Earth composition (> 85%
by mass composed of O, Mg, Si, Fe), similar Fe/Si and Mg/Si ratio and are carbon-poor.
It indicates formation processes similar to those controlling the formation and evolution of
objects in the inner solar system (Jura & Young 2014).
In our solar system, evidence suggests that rocky bodies were formed from chondriticlike materials (cf. Lodders & Fegley 2010). Current planet formation theory suggests that
the solar nebula was initially heated to very high temperatures to the extent that virtually everything was vaporized except for small amount of presolar grains (Ott 2007). The
– 10 –
nebula then cools to condense out various elements and mineral assemblages from the vapor phase at different temperatures according to the condensation sequence (White 2013).
Fe-Ni metal alloy and Mg-silicates condense out around similar temperatures of 1200-1400K
(depending on the pressure of the nebula gas) according to thermodynamic condensation calculation (Lodders 2003). Oxygen, on the other hand, does not have a narrow condensation
temperature range, as it is very abundant and it readily combines with all kinds of metals
to form oxides which condense out at various temperatures as well as H, N, C to form ices
condensing out at relatively low temperatures (Lewis 1997). As supported by the polluted
white dwarf study, we expect other exoplanetary systems to follow similar condensation sequence as the solar system in a H-dominated nebular environment for the major elements:
Fe, Mg, Si, and O (Jura & Young 2014).
In solar system, the primitive CI Carbonaceous Chondrites have Fe : Mg : Si = 0.855 :
1.047 : 1 (McDonough & Sun 1995). If all this Fe forms the core, CMF≈ 0.38 is the upper
limit. If some Fe is incorporated into the mantle either as metal or oxides, CMF becomes less.
The solar ratio of Fe/Si is representative for the stars in the solar neighborhood, which is a
tight distribution centered at 1, while the Mg/Si=1 seems to tend towards the lower end of
the distribution centered at 1.34 (Gilli et al. 2006; Grasset et al. 2009). A Mg/Si ratio higher
than 1 could produce more olivine (Mg2 SiO4 ) or more MgO to affect the mineralogy of the
upper mantle (Bond et al. 2010; Delgado Mena et al. 2010; Pagano et al. 2015). However, it
does not affect high-pressure EOS much. As we have pointed out earlier in Section 2 of EOS,
the TFD EOS of MgO, SiO2 , MgSiO3 , Mg2 SiO4 will converge above ∼1TPa due to their
identical average atomic weight of 20 and atomic charge of 10. Therefore, for more massive
planets the effect of Mg/Si tends to be smaller. The dispersion expected from the variation
in Mg/Si and Fe/Si ratios cause approximately 2% difference in radius (Grasset et al. 2009;
Dressing et al. 2015).
Oxygen is more readily available than Fe, Mg, or Si (Lodders 2003), as it is richly
produced in nuclear synthesis of massive stars and chemical evolution of Galaxy (Pagel
1997). So there is usually enough O to combine with Mg and Si to form Mg-silicates as well
as to oxidize some Fe-metal. Comparing bodies in our solar system: the core mass fractions
of Earth and Venus (Rubie et al. 2007) are around 0.3, the core mass fraction of Mars is
estimated to be 0.2 (McSween 2003), and of Vesta is smaller, about 0.17 (Ermakov et al.
2014; Ruzicka et al. 1997). For asteroid parent bodies of iron meteorites, their core mass
fractions are even smaller (Petaev & Jacobsen 2004). As such, there seems to exist a trend
of increasing CMF from smaller objects towards bigger objects in the solar system. This
trend seems to turn flat around 1 M⊕ .
These dense exoplanets between 2 and 5 M⊕ so far appear to agree with the mass-
– 11 –
radius relation with CMF ≈ 0.26, suggesting that they are like the Earth in terms of their
proportions of mantle and core. But their surface conditions are utterly different as they are
much too hot. This is due to observational bias that currently it is much easier for us to
detect close-in planets around stars. The fact that we now see so many of them suggests there
may be abundant Earth-like analogs at proper distances from their stars to allow existence
of liquid water on their surfaces.
4.
Summary
This paper provides a new CMF-dependent semi-empirical mass-radius relation for rocky
planets of 0≤CMF≤0.4 and 1M⊕ ≤ M ≤ 8M⊕ .
The result of fit to several dense exoplanets: CMF≈ 0.26 ± 0.07, agrees with the studies
of disintegrated planet debris in polluted white dwarf spectra, the solar system formation
theory and geochemical and cosmochemical evidence of meteorites.
The model tool is accessible at www.astrozeng.com.
5.
Acknowledgement
The author Li Zeng is grateful to Simons Foundation for supporting his postdoctoral
research position under the Simons Collaboration on the Origins of Life. The author Li
Zeng would like to thank Courtney Dressing, Lars Buchhave, and Eugenia Hyung for inspiring discussions. Part of this research was conducted under the Sandia Z Fundamental
Science Program and supported by the Department of Energy National Nuclear Security
Administration under Award Number DE-NA0001804 to S. B. Jacobsen (PI) with Harvard
University. This research is the authors’ views and not those of the DOE.
– 12 –
Fig. 3.— Mass-radius curves with planets color-coded by their surface temperatures (calculated from the stellar flux they receive assuming similar bond albedo as the Earth (≈0.3)
and perfect heat redistribution). (Pepe et al. 2013; Dumusque et al. 2014; Dressing et al.
2015; Ballard et al. 2013; Carter et al. 2012; Haywood et al. 2014; Barros et al. 2014;
Dragomir et al. 2013; Nelson et al. 2014; Winn et al. 2011; Vanderburg et al. 2015;
Cochran et al. 2011; Charbonneau et al. 2009; Motalebi et al. 2015; Berta-Thompson et al.
2015)
– 13 –
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